The existence, stability, and frequency of IP and AP states are wholly consistent
with a basic theory [1]} which models a pair of hydrodynamically coupled flagella as
beads rotating on springs with compliant radii \(R\)  separated by distance \(\ell \)  (see SI Text).
Assuming \(R\ll \ell \) , either IP or AP states of synchrony are predicted to be stable
depending on whether the beads are co- or respectively counter-rotating [1]}, [3]}.
Thus, CR-like configurations should tend to AP synchrony.
However in our experiments, flagella often come into such close proximity during certain phases of their beat cycles that
the far-field assumptions of the original model breakdown.
Non-local hydrodynamic interactions between different portions of flagella must now be considered for
the true flagellar geometry (rather than in a phase-reduced bead model).
In particular undulating filaments can be driven by fluid-structure interactions into either IP or AP oscillating modes
depending on initial relative phase [4]}.
The inherent stochasticity of flagellar beating [5]} thus leads to transitions between IP and AP states (Fig. REF ).
Hydrodynamic effects are notably stronger in the case of VC than CR, due to reduced screening by a smaller cell body and a distinctive
upward tilt of the flagellar beat envelope.
During evolution to multicellularity, this latter adjustment of BB orientation (Figs. REF C&D) facilitates beating of flagella confined within a spherical colony.
The CR mutant ptx1 also displays noisy transitions between IP and AP gaits [3]} due to an unknown mutation,
the implications of this we shall return to later (see Discussion).
